The Complexity of Cluster Vertex Splitting and Company

TL;DR

This paper studies overlapping clusterings through three lenses: Sigma Clique Cover (SCC), Cluster Vertex Splitting (CVS), and Cluster Editing With Vertex Splitting (CEVS). It establishes SCC as NP-hard via a Node Clique Cover reduction, then shows CVS is NP-hard by a tight SCC↔CVS correspondence using forward and backward lemmas, and finally derives a linear-vertex kernel for CVS. The work also corrects the critical-clique lemma, enabling a hardness proof for CEVS, which is then shown NP-hard through a CVS→CEVS reduction. Overall, the paper provides fundamental complexity results, a linear kernel, and a corrected theoretical tool with implications for overlapping clustering problems and related graph-editing tasks.

Abstract

Clustering a graph when the clusters can overlap can be seen from three different angles: We may look for cliques that cover the edges of the graph with bounded overlap, we may look to add or delete few edges to uncover the cluster structure, or we may split vertices to separate the clusters from each other. Splitting a vertex $v$ means to remove it and to add two new copies of $v$ and to make each previous neighbor of $v$ adjacent with at least one of the copies. In this work, we study underlying computational problems regarding the three angles to overlapping clusterings, in particular when the overlap is small. We show that the above-mentioned covering problem is NP-complete. We then make structural observations that show that the covering viewpoint and the vertex-splitting viewpoint are equivalent, yielding NP-hardness for the vertex-splitting problem. On the positive side, we show that splitting at most $k$ vertices to obtain a cluster graph has a problem kernel with $O(k)$ vertices. Finally, we observe that combining our hardness results with the so-called critical-clique lemma yields NP-hardness for Cluster Editing with Vertex Splitting, which was previously open (Abu-Khzam et al. [ISCO 2018]) and independently shown to be NP-hard by Arrighi et al. [IPEC 2023]. We observe that a previous version of the critical-clique lemma was flawed; a corrected version has appeared in the meantime on which our hardness result is based.

Figures (9)

• Figure 1: A graph with its unique minimum-weight sigma clique cover (left) and one of its multiple minimum-cardinality node clique covers (right).
• Figure 2: $K_3$ and $K_3^3$, illustrating \ref{['definition:universal_node']}.
• Figure 3: Example for our reduction from NCC to SCC. On the left, we see a NCC-instance, and on the right, we see the corresponding SCC-instance (only one universal node and its associated cliques are fully drawn). In both cases, a certificate is marked in the input graph, as well as stated explicitly.
• Figure 4: On the left, a graph $G$ with a sigma clique cover $\mathcal{C}$ is depicted. The clique $C_1 \in \mathcal{C}$ is marked in green. On the right, a graph $G'$, obtained by splitting $u$ into $u_\text{in}$ and $u_\text{out}$, is drawn. Additionally, a sigma clique cover $\mathcal{C}'$ of $G'$ is shown. The clique $C_1$ of $\mathcal{C}$ was "pulled away" to form $f(C_1)$ in the derived $\mathcal{C}'$, creating a sigma clique cover of "decreased overlap".
• Figure 5: A graph $G$ whose critical cliques are marked in blue (left) and $\mathop{\mathrm{CC}}\nolimits(G)$ (right).

Theorems & Definitions (35)

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